# MATHEMATICAL MODEL OF THE CURRENT TIME FOR THREE-FRAGMENT RADAR SIGNAL WITH NON-LINEAR FREQUENCY MODULATION

## Authors

• O. O. Kostyria Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine, Ukraine
• A. A. Нryzo Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine, Ukraine
• O. M. Dodukh Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine, Ukraine
• O. P. Narezhnyi V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, Ukraine
• A. V. Fedorov Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine, Ukraine

## Keywords:

radar signal; non-linear frequency modulation; autocorrelation function, side lobe level; mathematical model

## Abstract

Context. The authors of the article have developed a new mathematical model that allows taking into account frequency and phase distortions that occur in a three-fragment signal during the transition from one fragment to another, when the rate of frequency modulation of the signal changes. The object of research is the process of formation and processing of radar non-linear frequency modulation signals.

Objective. The purpose of the work is to develop and research a mathematical model of current time for a signal with non-linear frequency modulation, which consists of three linear frequency modulated fragments.

Method. The article provides a theoretical justification of the need to develop a mathematical model in the current time for a three-fragment signal with non-linear frequency modulation, capacity for work of the created model is demonstrated on the example of several radio signals that differ in frequency parameters. With the same signal parameters, the obtained results were compared with the results of the known model, for which known methods of spectral and correlation analysis were used. A distinctive feature of the proposed model is the consideration of jumps in the instantaneous frequency and phase of the signal that occur during the transition from one linear-frequency modulated fragment to the next. Such jump-like changes in frequency and phase in known models of signals with non-linear frequency modulation are not compensated for, which causes distortion of their spectra and an increase the side lobes level of auto-correlation (mutual-correlation) functions.

Results. A comparative check of the developed and known signal models indicates a decrease the side lobes level of the autocorrelation function by 3 dB or more, depending on the given frequency-time parameters.

Conclusions. The application of the proposed mathematical model makes it possible to form and process radar signals, which include three linear-frequency modulated fragments. Compensation of jump-like changes in frequency and phase leads to a decrease in the degree of distortion of the spectrum and, as a result, an increase in its effective width, which ensures a narrowing of the main lobe and a decrease the side lobes level of the auto-correlation function.

## Author Biographies

### O. O. Kostyria, Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine

Dr. Sc., Senior Research, Leading Research Scientist

### A. A. Нryzo, Ivan Kozhedub Kharkiv National Air Force University, Kharkiv, Ukraine

PhD, Associate Professor, Head of the Research Laboratory

### O. P. Narezhnyi, V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

PhD, Associate Professor

PhD, Research

## References

Skolnik M. I. Radar Handbook. Editor in Chief. Boston, McGraw-Hill Professional, second edition, 1990, 846 p.

Cook C. E., Bernfeld M. Radar Signals: An Introduction to Theory and Application. Boston, Artech House, 1993, 552 p.

Barton D. K. Radar System Analysis and Modeling. London, Artech House, 2005, 545 p.

Van Trees H. L. Detection, Estimation, and Modulation Theory. New York, John Wiley & Sons, 2004, 716 p.

Levanon N., Mozeson E. Radar Signals. New York, John Wiley & Sons, 2004, 403 p.

Melvin W. L., Scheer J. A. Principles of modern radar. New York, SciTech Publishing, 2013, 846 p.

Doerry A. W. Catalog of Window Taper Functions for Side lobe Control [Electronic resource]. Access mode: https://www.researchgate.net/publication/ 316281181_Catalog_of_Window_Taper_Functions_for_Sid elobe_Control.

Heinzel G., Rüdiger A., Schilling R. Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flattop windows [Electronic resource]. Access mode: https://pure.mpg.de/rest/items/item_ 52164_1/component/file_152163/content.

Galushko V. G. Performance Analysis of using tapered Windows for Side Lobe Reduction in Chirp-Pulse Compression, Radio Physics and Radio Astronomy, 2019, Vol. 24(4), pp. 300–313.

Nettem A. V., Daniel E. R., Chandu K. Windows For Reduction Of ACF Side Lobes of Pseudo-NLFM Signal, International Journal of Scientific & Technology Research, 2019, Vol. 8, Issue 10, pp. 2155–2161.

Ghavamirad R., Sebt M. A. Side Lobe Level Reduction in ACF of NLFM Waveform, IET Radar, Sonar & Navigation, 2019, Vol. 13, Issue 1, pp. 74–80.

Cook C. E. A class of nonlinear FM pulse compression signals, Proceedings of the IEEE, 1964, Vol. 52(11), pp. 1369– 1371.

Cook C. E., Paolillo J. A pulse compression predistortion function for efficient side lobe reduction in a high-power radar, Proceedings of the IEEE, 1964, Vol. 52(4), pp. 377– 389.

Xu Z., Wang X., Wang Y. Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications, Mathematics, 2022, Vol. 10, № 3939.

Fan Z., Meng H.-Y. Coded excitation with Nonlinear Frequency Modulation Carrier in Ultrasound Imaging System, IEEE Far East NDT New Technology & Application Forum (FENDT). Kunming, Yunnan province, China, 20–22 November 2020, pp. 31–35.

Chukka A., Krishna B. Peak Side Lobe Reduction analysis of NLFM and Improved NLFM Radar signal, AIUB Journal of Science and Engineering (AJSE), 2022, Vol. 21, Issue 2, pp. 125–131.

Saleh M., Omar S.-M., E. Grivel et al. A Variable Chirp Rate Stepped Frequency Linear Frequency Modulation Waveform Designed to Approximate Wideband Non-Linear Radar Waveforms, Digital Signal Processing, 2021, Vol. 109, № 102884.

Nettem A. V., Rani E. Doppler Effect Analysis of NLFM Signals, International Journal of Scientific & Technology Research, 2019, Vol. 8, Issue 11, pp. 1817–1821.

Kurdzo J. M., Cho John Y. N., Cheong B. L. et al. A Neural Network Approach for Waveform Generation and Selection with Multi-Mission Radar, IEEE Radar Conference. Boston, 22–26 April 2019, № 19043446.

Zhao Y., Ritchie M., Lu X. et al. Non-continuous piecewise nonlinear frequency modulation pulse with variable subpulse duration in a MIMO SAR Radar System, Remote Sensing Letters, 2020, Vol. 11, Issue 3, pp. 283–292.

Xu W., Zhang L., Fang C. et al. Staring Spotlight SAR with Nonlinear Frequency Modulation Signal and Azimuth NonUniform Sampling for Low Side Lobe Imaging, Sensors, 2021, Vol. 21, Issue 19, № 6487.

Song C., Wang Y., Jin G. et al. A Novel Jamming Method against SAR Using Nonlinear Frequency Modulation Waveform with Very High Side Lobes, Remote Sensing, 2022, Vol. 14, Issue 21, № 5370.

Jin G., Deng Y.-K., Wang R. et al. An Advanced Nonlinear Frequency Modulation Waveform for Radar Imaging With Low Side Lobe, IEEE Transactions on Geoscience and Remote Sensing, 2019, Vol. 57, Issue 8, pp. 6155–6168.

Valli N. A., Rani D. E., Kavitha C. Modified Radar Signal Model using NLFM, International Journal of Recent Technology and Engineering (IJRTE), 2019, Vol. 8, Issue 2S3, pp. 513–516.

Adithyavalli N. An Algorithm for Computing Side Lobe Values of a Designed NLFM function, International Journal of Advanced Trends in Computer Science and Engineering, 2019, Vol. 8, Issue 4, pp. 1026–1031.

Parwana S., Kumar S. Analysis of LFM and NLFM Radar Waveforms and their Performance Analysis, International Research Journal of Engineering and Technology (IRJET), 2015, Vol. 02, Issue 02, pp. 334–339.

Bayındır C. A Novel Nonlinear Frequency-Modulated Chirp Signal for Synthetic Aperture Radar and Sonar Imaging, Journal of Naval Science and Engineering, 2015, Vol. 11, Issue 1, pp. 68–81.

Jeyanthi J. E., Shenbagavalli A., Mani V.R.S. Study of Different Radar Waveform Generation Techniques for Automatic Air Target Recognition, International Journal of Engineering Technology Science and Research (IJETSR), 2017, Vol. 4, Issue 8, pp. 742–747.

Widyantara M. R., Suratman F. Y., Widodo S. et al. Analysis of Non Linear Frequency Modulation (NLFM) Waveforms for Pulse Compression Radar, Journal Electronica dan Telekomunikasi (JET), 2018, Vol. 18, №1, pp. 27–34.

Valli N. A., Rani D. E., Kavitha C. Performance Analysis of NLFM Signals with Doppler Effect and Back-ground Noise, International Journal of Engineering and Advanced Technology (IJEAT), 2020, Vol. 9, Issue 3, pp. 737–742.

Kostyria O. O., Hryzo A. A., Dodukh O. M. et al. Mathematical model of a two-fragment signal with a non-linear frequency modulation in the current period of time, Visnyk NTUU KPI Seriia – Radiotekhnika Radioaparatobuduvannia, 2023, Vol. 92, pp. 60–67.

2023-09-29

## How to Cite

Kostyria, O. O., Нryzo A. A., Dodukh, O. M., Narezhnyi, O. P., & Fedorov, A. V. (2023). MATHEMATICAL MODEL OF THE CURRENT TIME FOR THREE-FRAGMENT RADAR SIGNAL WITH NON-LINEAR FREQUENCY MODULATION . Radio Electronics, Computer Science, Control, (3), 17. https://doi.org/10.15588/1607-3274-2023-3-2